Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator
© Yin and Yang; licensee Springer. 2012
Received: 6 October 2011
Accepted: 7 March 2012
Published: 7 March 2012
The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)-Laplace operator with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri.
MSC: 35D05; 35J60; 58E05.
Keywordsp(x)-Laplacian Sobolev space three critical points theorem
where Ω ⊂ R N (N ≥ 2) is a bounded domain with boundary of class C1. ν is the outer unit normal to ∂Ω, λ, μ ≥ 0 are real numbers. with , N < q- ≤ q+, F : Ω × R × R → R is a function such that F(·, s, t) is measurable in Ω for all (s, t) ∈ R × R and F(x, ·, ·) is C1 in R × R for a.e. x ∈ Ω, F s denotes the partial derivative of F with respect to s. We assume G(x,s,t) and e p (x),e q (x) satisfy the following conditions:
e p (x),e q (x) ∈ L∞(Ω) and ess infΩ e p (x), ess infΩ e q (x) > 0, we denote ∥e p ∥1 = ∫Ω e p (x)dx and ∥e q ∥1 = ∫Ωe q (x)dx.
It is well known that the operator -Δp(x)= -div(|∇u|p(x)-2∇u) is called p(x)-Laplacian and the corresponding problem is called a variable exponent elliptic systems. The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been attracting attention of many authors in the last two decades. It arises from nonlinear elasticity theory, electro-rheological fluids, etc. see [1, 2], many results have been obtained on this kind of problems, for example [3–9]. For the special case, p(x) ≡ p(a constant), (1.1) becomes the well known p-Laplacian problem. There have been many papers on this class of problems, see [10–19] and the reference therein.
Recently, many papers have appeared in which the technical approach adopted is based on the three critical points theorem obtained by Ricceri . We cite papers [20–23], where the authors established the existence of at least three weak solutions to the problems with Dirichlet or Neumann boundary value conditions. Li and Tang in  obtained the existence of at least three weak solutions to problem (1) when p(x) ≡ p with Dirichlet boundary value conditions. El Manouni and Kbiri Alaoui  obtained the existence of at least three solutions of system (1) when p(x) ≡ p in Ω by the three critical points theorem obtained by Ricceri .
The main purpose of the present paper is to prove the existence of at least three solutions of problem (1). We study problem (1) by using the three critical points theorem by Ricceri  too. On the basis of , we state an equivalent formulation of the three critical points theorem in  as follows.
Theorem 1. Let X be a reflexive real Banach space, Φ : X → R a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous C1 functional, bounded on each bounded subset of X, whose Gâteaux derivative admits a continuous inverse on X*; Ψ : X → R a C1 functional with compact Gâteaux derivative. Assume that
(i) lim∥u∥→∞(Φ(u) + λ Ψ(u)) = ∞ for all λ > 0; and there are r ∈ R and u0, u1 ∈ X such that:
(ii) Φ(u0) < r < Φ(u1);
has at least three solutions in X whose norms are less than ρ.
The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper. In section 3, we establish our main result.
2 Notations and preliminaries
and (Lp(x)(Ω), | · |p(x)) becomes a Banach space, and we call it variable exponent Lebesgue space.
and we call it variable exponent Sobolev space. From , we know that spaces Lp(x)(Ω) and W1,p(x)(Ω) are separable, reflexive and uniform convex Banach spaces.
Then it is easy to see that is a norm on W1,p(x)(Ω) equivalent to ∥u∥p(x). In the following, we will use to instead of ∥ · ∥p(x)on W1,p(x)(Ω). Similarly, we use to instead of ∥ · ∥q(x)on W1,q(x)(Ω).
(i) |u|p(x)< 1(= 1; > 1) ⇔ ρ (u) < 1(= 1; > 1);
(iii) |u|p(x)→ 0(∞) ⇔ ρ (u) → 0(∞).
Proposition 3.If Ω ⊂ R N is a bounded domain, then the imbedding is compact whenever N < p-.
Proof. It is well know that is a continuous embedding, and the embedding is compact when N < p- and Ω is bounded. So we obtain the embedding is compact whenever N < p-.
Then X is a separable and reflexive Banach spaces. Naturally, we denote X* by the space (W1,p(x))*(Ω) × (W1,q(x))*(Ω), the dual space of X.
3 Existence of three solutions
Thus, we deduce that z ∈ X is a weak solution of (1) if z is a solution of (2). It follows that we can seek for weak solutions of (1) by applying Theorem 1.
We first give the following result.
Lemma 1. If Φ is defined in (6), then (Φ')-1 : X* → X exists and it is continuous.
for any z1 = (u1, v1), z2 = (u2, v2) ∈ X, i.e.,Φ' is uniformly monotone.
That's meaning Φ' is coercive on X.
By a standard argument, we know that Φ' is hemicontinuous. Therefore, the conclusion follows immediately by applying Theorem 26.A .
To obtain our main result, we assume the following conditions on F(x,s,t):
for a.e.x ∈ Ω and (s,t) ∈ R × R;
(A2) F(x,0,0) = 0 for a.e.x ∈ Ω;
Then we have the following main theorem.
Theorem 2. Assume (A1),(A2),(A3)(or (A3)'),(G) and (E) hold. Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ, there exists σ > 0 such that for each μ ∈ [0, σ], problem (1) has at least three weak solutions whose norms are less than ρ.
Proof. By the definitions of Φ, Ψ, J, we know that Ψ' is compact, Φ is weakly lower semi-continuous and bounded on each bounded subset of X. From lemma 1 we can see that (Φ')-1 is well defined, from condition (G), J is well defined and continuously Gâteaux differentiable on X, with compact derivative. Then we can use Theorem 1 to obtain the result. Now we show that the hypotheses of Theorem 1 are fulfilled.
and so the assumption (i) of Theorem 1 holds.
Now we consider in two cases:
Case (i): (A3) holds, i.e., there exist 1 ≤ |s1|, |t1| such that (9) hold.
Thus, (ii) of Theorem 1 is satisfied.
From (9)-(11) and the definition of r, we can see (iii) of Theorem 1 is hold.
So, (ii) of Theorem 1 is satisfied.
From (14) and (15), we can see (iii) of Theorem 1 is still hold.
Then all the hypotheses of Theorem 1 are fulfilled. By Theorem 1, we know that there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], problem (1) has at least three weak solutions whose norms are less than ρ.
By Theorem 2, we have the following result.
Corollary 1. Let f, g : Ω × R → R be Carathéodory functions, sup|ζ|≤s|g(·, ζ)| ∈ L1(Ω) for all s > 0, and define for any (x,t) ∈ Ω × R, e(x) ∈ L∞(Ω) and ess infΩe(x) > 0. Assume the following conditions hold.
for a.e.x ∈ Ω and t ∈ R;
has at least three weak solutions whose norms are less than ρ.
Remark 1. if p(x) = p in Ω, μ = 0, problem (17) was considered in . If we take f(x,t) = |t|γ(x)-2t - t with satisfies 2 < γ- ≤ γ+ < p-, μ = 0, Corollary 1 becomes a version of Theorem 2 in . Hence our Corollary 1 unifies and generalizes Theorem 2 in  and Theorem 2 in  and our Theorem 2 generalizes the main results of [21–25] to the system (1).
At last, we give two examples.
were are positive constants and c is given by (5). Then when M ≥ s1, F(x,u,v) defined in (18) satisfies (A1)-(A3) of Theorem 2, and G(x,u,v),e(x) satisfy
(G) and (E) respectively, by Theorem 2, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], system (19) has at least three weak solutions whose norms are less than ρ.
i.e., (A3)' hold for F(x,u,v) defined in (22).
has at least three weak solutions whose norms are less than ρ.
Remark 2. We remark that the methods used in this paper are also applicable for the cases of the other boundary value conditions, for example, Dirichlet boundary value conditions.
The project supported by the National Natural Science Foundation of China (No. 11171092). Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005).
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